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Abstract

We present a formulation of optical point spread function based on a scaled three-dimensional Fourier transform expression of focal field distribution and the expansion of generalized aperture function. It provides an equivalent but more flexible representation compared with the analytic expression of the extended Nijboer-Zernike approach. A phase diversity algorithm combined with an appropriate regularization strategy is derived and analyzed to demonstrate the effectiveness of the presented formulation for phase retrieval and deconvolution. Experimental results validate the performance of presented algorithm.

Figures (8)

Simulation results for the two-dimensional phase retrieval (all units are the wavelength), where (a-c) are the cases of circular aperture and (d-f) are the cases of annular aperture. (a) and (d) are the randomly generated phase aberrations with the same RMS = 0.222λ. (b) and (e) are the reconstructed phases with amplitude regularization, where the RMS of residual errors are 0.023λ and 0.032λ, respectively. (c) and (f) are the reconstructed phases without amplitude regularization, where the RMS of residual errors are 0.031λ and 0.068λ, respectively.

Simulation results for the three-dimensional deconvolution, where (a) is the stack of the simulated object, (b) is the stack of the focused image, and (c) is the stack of the reconstructed object. The RMS of randomly generated phase aberrations is 0.182λ, and the RMS of residual error of reconstructed phase is 0.07λ.

Zernike coefficients for a generalized aperture function with uniform amplitude and large phase aberration of RMS = 1.398λ, where the charts from top to bottom are the α coefficients and the corresponding imaginary and real parts of β coefficients, respectively.

Experimental results of phase retrieval and deconvolution for the phase aberration of RMS = 0.647λ. (a) and (d) are the measured and reconstructed phases respectively, where the RMS of residual error is 0.046λ. (b) and (e) are the measured and reconstructed focal PSF, respectively. (c) and (f) are the simulated focal image and reconstructed object, respectively.

Experimental results of phase retrieval and deconvolution for the phase aberration of RMS = 1.203λ. (a) and (d) are the measured and reconstructed phases respectively, where the RMS of residual error is 0.202λ. (b) and (e) are the measured and reconstructed focal PSF, respectively. (c) and (f) are the simulated focal image and reconstructed object, respectively.